Space-Time Block Preconditioning for Incompressible Flow

Journal Article · 08 February 2022 · SIAM Journal on Scientific Computing

DOI: https://doi.org/10.1137/21m1390773 · OSTI ID:1864996

Danieli, Federico  ^[1]; Southworth, Ben Scott  ^[2]; Wathen, Andrew J. ^[1]

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1. Univ. of Oxford (United Kingdom)
2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Parallel-in-time methods have become increasingly popular in the simulation of time-dependent numerical PDEs, allowing for the efficient use of additional message passing interface processes when spatial parallelism saturates. Most methods treat the solution and parallelism in space and time separately. In contrast, all-at-once methods solve the full space-time system directly, largely treating time as simply another spatial dimension. All-at-once methods offer a number of benefits over separate treatment of space and time, most notably significantly increased parallelism and faster time to solution (when applicable). However, the development of fast, scalable all-at-once methods has largely been limited to time-dependent (advection-)diffusion problems. This paper introduces the concept of space-time block preconditioning for the all-at-once solution of incompressible flow. By extending well-known concepts of spatial block preconditioning to the space-time setting, we develop a block preconditioner whose application requires the solution of a space-time (advection-)diffusion equation in the velocity block, coupled with a pressure Schur complement approximation consisting of independent spatial solves at each time-step, and a space-time matrix-vector multiplication. The new method is tested on four classical models in incompressible flow. Finally, the results indicate perfect scalability in refinement of spatial and temporal mesh spacing, perfect scalability in nonlinear Picard iteration count when applied to a nonlinear Navier--Stokes problem, and minimal overhead in terms of number of preconditioner applications compared with sequential time-stepping.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: Engineering and Physical Sciences Research Council (EPSRC); USDOE
• Grant/Contract Number: 89233218CNA000001
• OSTI ID: 1864996
• Report Number(s): LA-UR-21-20115
• Journal Information: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 1 Vol. 44; ISSN 1064-8275
• Publisher: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement
• Country of Publication: United States
• Language: English